You know that the paper is a model for a plane surface. When you join a number of points without lifting a pencil from the paper (and without retracing any portion of the drawing other than single points), you get a plane curve. Try to recall different varieties of curves you have seen in the earlier classes. Match the following: (Caution! A figure may match to more than one type).
A simple closed curve made up of only line segments is called a polygon.
Curves Try to give a few more examples and non-examples for a polygon.
Draw a rough figure of a polygon and identify its sides and vertices.
3.2.1 Classification of polygons We classify polygons according to the number of sides (or vertices) they have.
3.2.2 Diagonals A diagonal is a line segment connecting two non-consecutive vertices of a polygon
3.2.3 Convex and concave polygons Here are some convex polygons and some concave polygons.
3.2.4 Regular and irregular polygons A regular polygon is both ‘equiangular’ and ‘equilateral’. For example, a square has sides of equal length and angles of equal measure. Hence it is a regular polygon. A rectangle is equiangular but not equilateral. Is a rectangle a regular polygon? Is an equilateral triangle a regular polygon? Why?
3.2.5 Angle sum property Do you remember the angle-sum property of a triangle? The sum of the measures of the three angles of a triangle is 180°. Recall the methods by which we tried to visualise this fact. We now extend these ideas to a quadrilateral.
On many occasions a knowledge of exterior angles may throw light on the nature of interior angles and sides.This is true whatever be the number of sides of the polygon. Therefore, the sum of the measures of the external angles of any polygon is 360°.
Based on the nature of the sides or angles of a quadrilateral, it gets special names.
3.4.1 Trapezium Trapezium is a quadrilateral with a pair of parallel sides.
(1)Take four set-squares from your and your friend’s instrument boxes. Use different numbers of them to place side-by-side and obtain different trapeziums. If the non-parallel sides of a trapezium are of equal length, we call it an isosceles trapezium. Did you get an isoceles trapezium in any of your investigations given above?
(2)3.4.2 Kite Kite is a special type of a quadrilateral. The sides with the same markings in each figure are equal. For example AB = AD and BC = CD.
(3)3.4.3 Parallelogram A parallelogram is a quadrilateral. As the name suggests, it has something to do with parallel lines.
We obtain a Rhombus (which, you will see, is a parallelogram) as a special case of kite
(which is not a a parallelogram).
3.5.2 A rectangle A rectangle is a parallelogram with equal angles (Fig 3.37). What is the full meaning of this definition? Discuss with your friends. If the rectangle is to be equiangular, what could be the measure of each angle?